As introduced above, a key component of the metal circulation is turbulence, either generated by the AGN feedback or by the large-scale cosmological evolution (§REF ), which is worth to further dissect. Remarkably, turbulent motions generate two (seemingly) contrasting effects, but on different scales. On the one hand, turbulent motions promote the diffusion of metals, tending to equalize the radial abundance profile from a negative to null gradient [1]}, [2]}. On the other hand, turbulence induces local chaotic relative density fluctuations \(\delta \rho /\rho \) . As per above Eq. REF , metals can be considered on average akin to passive tracers of the HD density field, thus \(\delta \rho _Z/\rho _Z\sim \delta \rho /\rho \) . As shown by numerical and analytic studies [3]}, [4]}, such stratified hot-halo fluctuations are linearly tied to the turbulent Mach number \({\rm Ma_t}\) , hence the relative metal abundance can help us to constrain the level of turbulence in the IGrM too, \(\delta \rho _Z/\rho _Z \propto {\rm Ma_t}\) , with the slope of the Fourier spectrum constraining plasma processes such as thermal conduction [5]}. In the IGrM, the inferred 3D sonic Mach number of turbulence is \({\rm Ma_t} \sim 0.3-0.5\)  [6]}, [7]}, i.e., \(\sigma _v\)  of a few 100 \({\rm km\,s^{-1}}\) .
This can complement upcoming spectral X-ray IFU studies carried out via XRISM and Athena (see §), with detailed synthetic observations already highlighting unprecedented features of metals and turbulence in hot halos [8]}, [9]}, [10]}, [11]}. Moreover, constraining the turbulent metal evolution in the IGrM plasma phase enables to assess the kinematics of the top-down multiphase rain, since the condensed warm (H\(\alpha \) +[NII]) filaments and cold (CO, HI) clouds share analogous ensemble velocity dispersion [12]}, [13]}, [14]}, [15]}.
